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(MATH244)midterm00F.pdf
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Mid-term Examination
MATH 244 (L2) Applied Statistics Mid-Term Examination October 20, 2000
Name ___________________ Student ID ______________ Tutorial section ____
Time allowed : 1 hour Answer all questions.
1. (10%) Circle the appropriate answers for the following true or false questions. One mark will be given for each correct answer. Half mark will be deducted for each incorrect answer.
(a)
Let A be an event with 0 <Pr A T / F

()<1. It is impossible that the event A and A are independent .

(b)
If A and B are independent events, A and B are independent too. T / F

(c)
A group of events are called to be mutually exclusive if any two of them are T / F independent to each other.

(d)
For any symmetric dataset, sample mean = sample median = mode. T / F

(e)
If A is an event with Pr A T / F

()=1, then A must be equal to the sample space.

(f)
Sample mean is always the best measure of location since it uses all the data. T / F

(g)
If the distribution of a set of data is skewed to the right, then the median T / F should be closer to the upper quartile than to the lower quartile.

(h)
A stem-and-leaf plot is more informative than a histogram. T / F

(i)
If the moment generating function of a random variable X is T / F



t
MX ()t =p
[1.e (1.p)], t <.ln(1.p),
then X will have the memoryless property.

(j) Histogram is not a suitable form of representation of frequency distribution T / F table when the class-intervals have unequal widths.
P. 1
Mid-term Examination
2. (10%) Suppose John try to answer all the ten questions in problem 1 by pure guess. Let X be the number of correct answers and Y be the marks he can obtain in problem 1.
(a)
What is the distribution of X ? Write down the corresponding assumptions.

(b)
Find the expected value and variance of X .

(c)
Express Y in terms of X.

(d)
Use the result in (c), or otherwise, to find the expected value and variance of Y.

(e)
What is the probability that John will get negative marks from problem 1?

(f)
What is the probability that John will give the same set of answers as you in problem 1?



Solution :
(a) X ~ b(10,0.5)
Assumptions : For each question, John will answer true with probability 0.5. His choices on the answers of all questions are independent.
EX np ()10 0.5
() ( )()=np .=10 ()0.5 0.5 =

(b) ()== ()=5
Var X 1 p ()2.5
(c) Y =X +(10 .X )...1 .. (1 mark for correct answer, -0.5 mark for incorrect answer)
.2 .
3

= X .5
2
(d) ()3 E()X .5 = 3 ()5
EY = 5 .=2.5
22
32 45
.Var X 5.625
Var()Y =... ()= =
.2 . 8
3 10
(e) From (c), Y<0 . X .5 <0 .X <.X 3
23
Hence Pr( )( =Pr X

Y <0 3)
.10. 0 10 .10. 19 .10. 28 .10. 37



()0.5 () + 0.5 () ()0.5 ()+ 0.5 ()
= 0.5 ()0.5 + 0.5 ()0.5
.
.
.
.
.
.
.
.

0 123
.. ...... =0.171875
(f) Probability that John will give the same set of answers as me in problem 1
=()1